The Time Domain Integral Equation method for electromagnetics is an appealing
computational method for many applications in industry. However, its applicability has long been suffering from instabilities. A rigorous analysis of the variational formulation is imperative to the successful design of stable and robust numerical schemes. In this paper, an established functional framework and stability theorem will be extended to the differentiated version of the electric field integral equations, which can be discretized more efficient and is more often used in engineering literature. The extended stability theorem, combined with efficiency requirements, will give guidelines on the choice of
test and basis functions of the space–time Petrov–Galerkin scheme. A discrete equivalence with the collocation method results in the recommendation to choose the quadratic spline basis function in the standard Marching-on-in-Time scheme. Computational experiments confirm that the quadratic spline basis functions have superior stability characteristics compared to the conventional quadratic Lagrange basis functions in time.
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